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Question
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.
Options
2
`(-1)/(2sqrt(1 - x^2)`
`2/x`
1 – x2
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Solution
The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is 2.
Explanation:
Let y = cos–1(2x2 – 1) and t = cos–1x
Differentiating both the functions w.r.t. x
`"dy"/"dx" = "d"/"dx" cos^-1 (2x^2 - 1)` and `"dt"/"dx" = "d"/"dx" cos^-1x`
⇒ `"dy"/"dx" = (-1)/sqrt(1 - (2x^2 - 1)^2) * "d"/"dx" (2x^2 - 1)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`
= `(-1.4x)/sqrt(1 - (4x^4 + 1 - 4x^2)` and `"dt"/"dx" = (-1)/sqrt(1 - x^2)`
= `(-4x)/sqrt(1 - 4x^4 - 1 + 4x^2)`
= `(-4x)/sqrt(4x^2 - 4x^4)`
= `(-4x)/(2xsqrt(1 - x^2)`
⇒ `"dy"/"dx" = (-2)/sqrt(1 - x^2)`
Now `"dy"/"dx" = ("dy"/"dx")/("dt"/"dx")`
= `((-2)/sqrt(1 - x^2))/((-1)/sqrt(1 - x^2))`
= 2.
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